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A077478
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Rectangular array R read by antidiagonals: R(i,j) is the number of integers k that divide both i and j (i >= 1, j >= 1).
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4
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1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 3, 1, 2, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 2, 1, 2, 2, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 1, 4, 1, 3, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 2, 2, 1, 1, 2, 2, 1, 2, 1, 1
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OFFSET
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1,5
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COMMENTS
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Antidiagonal sums of R, alias row sums of T, are essentially A065608. Diagonal elements of R comprise A000203 (sums of divisors of n).
If R(n) is the n X n Redheffer matrix (A143104) and Rt(n) is its transposed matrix, then this sequence seems to be formed by R(n)*Rt(n). - Enrique Pérez Herrero, Feb 21 2012
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LINKS
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FORMULA
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R=U*V, where U and V are the summatory matrices (A077049, A077051). The triangle T(n, k) formed by antidiagonals: T(n, k)=tau(gcd(k, n+1-k)) for 1<=k<=n, where tau(m)=A000005(m). [Corrected by Leroy Quet, Apr 08 2009]
Dirichlet g.f.: Sum_{n>=1} Sum_{k>=1} tau(gcd(n,k))/n^s/k^c = zeta(s)*zeta(c)* zeta(s + c). - Mats Granvik, May 19 2021
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EXAMPLE
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First few rows of the array R are:
1, 1, 1, 1, 1, 1, 1, ...
1, 2, 1, 2, 1, 2, 1, ...
1, 1, 2, 1, 1, 2, 1, ...
1, 2, 1, 3, 1, 2, 1, ...
1, 1, 1, 1, 2, 1, 1, ...
1, 2, 2, 2, 1, 4, 1, ...
...
First few rows of the triangle T are:
1;
1, 1;
1, 2, 1;
1, 1, 1, 1;
1, 2, 2, 2, 1;
1, 1, 1, 1, 1, 1;
1, 2, 1, 3, 1, 3, 1;
1, 1, 2, 1, 1, 2, 1, 1;
1, 2, 1, 2, 2, 2, 1, 2, 1;
1, 1, 1, 1, 1, 1, 1, 1, 1, 1;
1, 2, 2, 3, 1, 4, 1, 3, 2, 2, 1;
...
R(4,2)=2 since 1|2, 1|4 and 2|2, 2|4.
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MATHEMATICA
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T[n_, k_]:=DivisorSigma[0, GCD[n, k]]; Flatten[Table[T[n-k+1, k], {n, 14}, {k, n}]] (* Stefano Spezia, May 23 2021 *)
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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